The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI-modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of d-step central stability, and prove that if the ideal of relations of a category is generated in degrees at most d, then every module presented in finite degrees is d-step centrally stable.

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.

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We give two homological proofs of the Noetherianity of the category FI, a fundamental result discovered by Church, Ellenberg, Farb, and Nagpal.

We show that the standard graded cover of the well-known category $\mathcal{O}$ of Bernstein–Gelfand–Gelfand can be characterized by its compatibility with the action of the center of the enveloping algebra.